$(4)(25$pts$)$ Evaluate ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{x^2}{(3+4x-4x^2{)}^{\frac{1}{2}}}dx$

Step $1$: Fill out the box: $(3+4x-4x^2{)}^{\frac{3}{2}}=(\sqrt[\phantom{2}]{3+4x-4x^2}{)}^{}$;$3$

Step $2$: $\sqrt[\phantom{2}]{3+4x-4x^2}=\sqrt[\phantom{2}]{4-(4x^2-4x+1)}=\sqrt[\phantom{2}]{4-(2x-1{)}^2},$ wr and the new integral;